We have adopted the definition of complexity as suggested by Herrera
[1]}), with the intention of extending this
to the \(f(\mathcal {G}, \mathbf {T}^{2})\)  model of gravity for
dynamical gravitational compact objects evolving quasi-homologously.
We initiate by formulating the \(f(\mathcal {G}, \mathbf {T}^{2})\) 
field equations via \(\mathrm {EH}\) -action. Then we formulate the
geometric mass \(m\)  via Misner-Sharp methodology and developed an
explicit relationship between structural variables, mass function,
and the Weyl scalar. The curvature tensor is then divided
orthogonally to procure a dynamical variable \(Y_{TF}\)  which is
connected to the structural characteristics (i.e., anisotropic
stresses, \(f(\mathcal {G}, \mathbf {T}^{2})\)  corrections and the
density inhomogeneity) of the spherical gravitational system. The
above-stated variables are the primary ingredients for provoking
complexities in any gravitational compact system. The above-stated
variables are the primary ingredients for provoking complexities in
any gravitational compact system. The presence of these variables in
the dynamical variable \(Y_{TF}\)  is a fundamental cause of entitling
this as the \(\mathbb {C}_{\mathbf {F}}\) . It is observed that the
presence of higher-order matter stresses along with the Gauss-Bonnet
curvature terms increases the complexity of the gravitational
source. Some of the significant changes as a consequence of
\(\mathbb {C}_{\mathbf {F}}\)  are described as
